The AJ-conjecture and cabled knots over the figure eight knot

TL;DR

This paper demonstrates that most cabled knots over the figure eight knot in three-dimensional space satisfy the AJ-conjecture, especially when the cabling parameters meet certain numerical conditions.

Contribution

It proves the AJ-conjecture for a broad class of cabled knots over the figure eight knot, extending previous results in knot theory.

Findings

Most cabled knots over the figure eight knot satisfy the AJ-conjecture.

The AJ-conjecture holds for any $(r,s)$-cabled knot over the figure eight knot if $r$ is outside the interval $[-4s, 4s]$.

The result applies to a wide range of cabled knots, confirming the conjecture in these cases.

Abstract

We show that most cabled knots over the figure eight knot in $S^3$ satisfy the AJ-conjecture, in particular, any $(r,s)$-cabled knot over the figure eight knot satisfies the $AJ$-conjecture if $r$ is not a number between $-4s$ and $4s$.